Data from a dedicated cosmic ray run of the ALEPH detector were used in a study of muon trident production, i.e., muon pairs produced by muons. Here the overburden and the calorimeters are the target materials while the ALEPH time projection chamber provides the momentum measurements. A theoretical estimate of the muon trident cross section is obtained by developing a Monte Carlo simulation for muon propagation in the overburden and the detector. Two muon trident candidates were found to match the expected theoretical pattern. The observed production rate implies that the nuclear form factor cannot be neglected for muon tridents.